# Analytic Geometric Trigonometry

#### Hartlw

Any angle can be assigned a rational number by dividing a standard angle, say 90deg, into an arbitrary number of segments geometrically, just as points on the line can.
Then any angle can be defined (given a number) by a geometric cut in precisely the same way that any point on the line can.

Any angle in radians is pi times the angle in degrees, and arc length follows accordingly. The rational geometric definirion of angles by rational numbers (degrees), which is more fundamental than radians, was done by the Sumerians 5000 years ago.

With any angle defined geometrically for all x, trigonometric functions are defined geometrically by the standard definitions for all x, and can be treated analytically without series.

And note that now the standard geometric proof of sinx/x, using geometry and the squeeze theorem, is valid.

I note in passing that with x and sinx defined analytically without series, SlipEternal’s proof of (x-sinx)/x3 < 16, in http://mathhelpforum.com/calculus/229484-challenging-limit.html , post#7, which I am locked out of, is correct, but if I were a teacher I wouldn’t give him credit because he didn’t define sinx analytically without series, which in all fairness, he couldn’t have done without this post.

#### Hartlw

To avoid the abstract notion of a cut, you can assign a decimal to an arbitrary angle geometrically “just” as Kaplan assigns a decimal geometrically to an arbitrary point on a line:

Pick a point on the line.
Divide the line into 10 segments (HS geometry).
Pick the segment which contains the line and divide that into ten segments.
Continue indefinitely or until the point falls on the end of a segment.

You can do the same thing with an angle with an interesting caveat. You can’t divide an angle into 10 segments using HS geometry but you can bisect it. So let 90deg = 1 and pick an arbitrary angle. Bisect angle 1 and pick the segment which contains the angle. Then bisect that segment. Continue indefinitely or until your angle falls on the edge of a segment. You then have a binary representation of any angle which you can convert to a decimal.

#### Prove It

MHF Helper
Any angle can be assigned a rational number by dividing a standard angle, say 90deg, into an arbitrary number of segments geometrically, just as points on the line can.
Then any angle can be defined (given a number) by a geometric cut in precisely the same way that any point on the line can.
There is so much wrong with this statement. First of all, the set of rational numbers is not continuous, which means that it would be impossible to define the sine function as a continuous function.

Also, not all points on the line can be expressed as a ratio of two known lengths. The standard counterexample is the diagonals of a unit square, of length \displaystyle \begin{align*} \sqrt{2} \end{align*}.

Any angle in radians is pi times the angle in degrees,
No it's not. Actually any angle in radians is \displaystyle \begin{align*} \frac{\pi}{180} \end{align*} times the angle in degrees. Surely since the radian measure is defined as the number of lengths of the radius swept out on the circumference, and the circumference is equal to \displaystyle \begin{align*} 2\pi \cdot \,\textrm{radius} \end{align*}, there are \displaystyle \begin{align*} 2\pi \end{align*} radians in a circle, and thus the ratio of degrees to radians is:

\displaystyle \begin{align*} 360^{\circ} &: 2\pi ^{C} \\ 1^{\circ} &: \frac{2\pi^C}{360} \\ 1^{\circ} &: \frac{\pi^C}{180} \end{align*}

The rational geometric definirion of angles by rational numbers (degrees), which is more fundamental than radians, was done by the Sumerians 5000 years ago.
Which was before the time when the Pythagoreans realised the existence of irrational numbers.

I also have a problem with you saying degrees are more fundamental than radians. Radian measurement is the fundamental measurement, it being a continuous DIMENSIONLESS amount, making it logical to substitute into functions (after all, a dimensionless quantity is just a number). Also, nearly all mathematics involving trigonometric functions involves noting some sort of oscillation, which can only happen when you are allowed to continually go around the circle in both directions as many times as necessary, as opposed to degrees, which by definition ARE a ratio, and so making any "negative angle" or "angle greater than \displaystyle \begin{align*} 360^{\circ} \end{align*}" meaningless.

With any angle defined geometrically for all x, trigonometric functions are defined geometrically by the standard definitions for all x, and can be treated analytically without series.
I agree with this statement. And the radian measure is defined geometrically as the number of lengths of the radius on the circumference.

And note that now the standard geometric proof of sinx/x, using geometry and the squeeze theorem, is valid.
As it always has been without your nit-picking...

I note in passing that with x and sinx defined analytically without series, SlipEternal’s proof of (x-sinx)/x3 < 16, in http://mathhelpforum.com/calculus/229484-challenging-limit.html , post#7, which I am locked out of, is correct, but if I were a teacher I wouldn’t give him credit because he didn’t define sinx analytically without series, which in all fairness, he couldn’t have done without this post.
I have a real issue with this statement. In mathematics, it is expected that things are "well known", such as the geometric definitions of the trigonometric functions. It is IMPOSSIBLE to prove every statement needed in order to make a mathematical argument. To not give any credit would be to say that all the hard work that the student has put in would be meaningless. This is counter-productive. The hard work and attempts should be worth something, as it is the hard work which will ensure success down the road. Besides, everything that was needed was there anyway.

#### Hartlw

Def: Angle x in radians = 2π/360 times angle a in degrees.
Theorem: Arclength = rx, r= radius of arc.
Proof: Circumference of a circle = 2πr
Arclength = (a/360)2πr = rx

This also corrects the obvious typo in OP, and makes the important point that arclength = xr, is a theorem, not the customary meaningless (circular) definition for r==1.

I tend to be a little concise sometimes, any rational number m/n can be defined by dividing a line or angle into n parts and taking m of them. I would have thought that was obvious. The OP is self-explanatory. Others are entitled to their opinion. I do not wish to get into another post-fest unless someone has a really serious objection.

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#### romsek

MHF Helper
Def: Angle x in radians = 2π/360 times angle a in degrees.
Theorem: Arclength = rx, r= radius of arc.
Proof: Circumference of a circle = 2πr
Arclength = (a/360)2πr = rx

This also corrects the obvious typo in OP, and makes the important point that arclength = xr is a theorem, not the customary meaningless (circular) definition.

I tend to be a little concise sometimes, any rational number m/n can be defined by dividing a line or angle into n parts and taking m of them. I would have thought that was obvious. The OP is self-explanatory. Others are entitled to their opinion. I do not wish to get into another post-fest unless someone has a really serious objection.
The main problem with the OP is that you state any angle can be represented by a rational number which is clearly untrue.

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#### Hartlw

The main problem with the OP is that you state any angle can be represented by a rational number which is clearly untrue.
Thanks. That is a serious objection. And in one sentence yet. I realized this in post 2, but failed to draw the obvious conclusion for post 1. The procedure of post 2, sequential bi-section, geometrically assigns to every angle a rational or irrational number, just as Kaplan's divide-by-ten procedure does for a line. Without it, "let x be the distance along a line" is a meaningless statement because it has no prescription for assigninng a number to x, which makes it analytically meaningless.

#### Hartlw

Cutting the Gordian Knot

Definition: Arc Length:
Draw a circle of radius 1.
Attach a string of any length between 0 and pi/2 (or pi, ..).
Wrap it around the circle and mark end point.

Definition: angle x = arc length.

Now lim sinx/x can be treated analytically geometrically.

This should be at the beginning of every calculus book for Engineers and Scientists, instead of x=arc length: What is x, arc length. What is arc length, x.

I note that in another post the question what is sinx (excluding series) received no response.

#### Hartlw

The beauty of the “geometric”* construction in previous post is that you don’t even have to know what pi is, ie, you don’t need a series for it (note challenge problem).

Attach a string to a point on the unit circle and wrap it around to the opposite point and mark it. That is the length pi which defines any length on the line, and hence the length of any arc on the circle by above construction (“geometry”).

*It’s not straight edge and compass (the Gordian Knot), but we have to move on.